The P-property conjecture under the conditions of Theorem main2

Let $(X,p)$ be a partial $b$-metric space, and let $T:X\to X$ be a self-map satisfying the conditions of Theorem main2. Recall that $T$ has the $P$ property if

$$F(T)=F(T^n)$$

for every $n\in\mathbb{N}$, where $F(T)$ denotes the fixed-point set of $T$.

P-property conjecture. Under the conditions of Theorem main2, $T$ has the $P$ property. The source further says that it is enough to check whether $T$ satisfies the condition

$$p(Tx,T^2)\leq\lambda p(x,Tx).$$

This is presented as an unproved conjecture concerning the fixed points of iterates of mappings on partial $b$-metric spaces. The precise conditions of Theorem main2 are not included in the supplied text and should be checked in the source.